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Multiplicity of 1D-concentrated positive solutions to the Dirichlet problem for an equation with \(p\)-Laplacian. (English. Russian original) Zbl 1326.35162

Funct. Anal. Appl. 49, No. 2, 151-154 (2015); translation from Funkts. Anal. Prilozh. 49, No. 2, 88-92 (2015).
Summary: We consider the Dirichlet problem for the equation \(-\Delta_p = u^{q-1}\) with \(p\)-Laplacian in a thin spherical annulus in \(\mathbb{R}^n\) with \(1 < p < q < p^\ast_{n-1}\), where \(p^\ast_{n-1}\) is the critical Sobolev exponent for embedding in \(\mathbb{R}^{n-1}\) and either \(n = 4\) or \(n \geqslant 6\). We prove that this problem has a countable set of solutions concentrated in neighborhoods of certain curves. Any two such solutions are nonequivalent if the annulus is thin enough. As a corollary, we prove that the considered problem has as many solutions as required, provided that the annulus is thin enough.

MSC:

35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35B09 Positive solutions to PDEs
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References:

[1] Byeon, J., No article title, J. Differential Equations, 136, 136-165 (1997) · Zbl 0878.35043
[2] Coffman, C. V., No article title, J. Differential Equations, 54, 429-437 (1984) · Zbl 0569.35033
[3] Kolonitskii, S. B., No article title, Algebra i Analiz, 22, 206-221 (2010)
[4] Li, Y. Y., No article title, J. Differential Equations, 83, 348-367 (1990) · Zbl 0748.35013
[5] Malchiodi, A., No article title, Boll. Unione Mat. Ital., 8, 615-628 (2005) · Zbl 1182.35121
[6] Mizoguchi, N.; Suzuki, T., No article title, Houston J. Math., 22, 199-215 (1996) · Zbl 0862.35036
[7] Nazarov, A. I., No article title, Proc. St.-Petersburg Math. Soc., 10, 33-62 (2004)
[8] Nazarov, A. I., No article title, Probl. Math. Anal., 20, 171-190 (2000) · Zbl 1002.46025
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